ABSTRACT
The Dirichlet problem is very important in the study of harmonic functions and conformal mapping. Equally important is the classical Neumann problem. Suppose Ω is a bounded domain with C∞ smooth boundary. Given a function ψ in C∞(bΩ), the Neumann problem is to find a function ϕ ∈ C∞(Ω) that is harmonic on Ω such that the normal derivative of ϕ on the boundary is equal to ψ. We will use the notation ∂ϕ/∂n to denote the normal derivative of ϕ. The first observation to be made is that not every ψ can be equal to the normal derivative of a harmonic function. Indeed, if ϕ is harmonic, then Gauss’ theorem yields∫
(∂ϕ/∂n) ds =
∫∫ Ω
∆ϕ dA = 0.
